Application of group analysis to classification of systems of three second‐order ordinary differential equations

Abstract

Here, we give a complete group classification of the general case of linear systems of three second‐order ordinary differential equations excluding the case of systems which are studied in the literature. This is given as the initial step in the study of nonlinear systems of three second‐order ordinary differential equations. In addition, the complete group classification of a system of three linear second‐order ordinary differential equations is carried out. Four cases of linear systems of equations are obtained. Copyright © 2015 John Wiley & Sons, Ltd.

Equivalence groupoids of classes of linear ordinary differential equations and their group classification

“…Only recently the group classification of systems of second-order linear ODEs with commuting constant-coefficient matrices was considered for various particular cases of the number of equations (two, three or four) and of the structure of the coefficient matrices in a series of papers [5,6,27,43] and was then exhaustively solved in [4]. In spite of a number of publications on the subject, the group classification of systems of linear second-order ODEs with noncommuting constant-coefficient matrices or with general nonconstant coefficients was carried out only for the cases of two and three equations [28,40,44]. The consideration of a greater number of equations or equations of higher and different orders within the framework of the standard "compatibility" approach requires cumbersome computations.…”

On group classification of normal systems of linear second-order ordinary differential equations

“…This result has been stated as a Theorem in Section 5 of the paper. The results were discussed and in two or more of the cases which had the admitted generators of the form X A the study was similar to that done in the earlier studies in [1,2,3,4]. A detailed discussion on systems admitting generators of the form X A has been given in Section 6 of the paper.…”

“…This has been part of the reason for focusing on such systems in the current study. We study the general group classification of systems of linear secondorder ordinary differential equations motivated by recent results obtained in [1,2,3,4]. Linear equations play an important role in many applications where they occur in a disguised form.…”

Integrability of systems of two second-order ordinary differential equations admitting four-dimensional Lie algebras

“…In particular, group classification of systems of two second-order ODEs admitting three-and fourdimensional Lie algebras is provided in [19,20]. Group classification of systems of three secondorder ODEs is provided in [21]. Lie symmetries of systems of second-order linear ODEs are investigated in [22][23][24][25].…”